A Pfaffian formula for the Ising partition function of surface graphs

preprint2020arXivOpen access

Abstract

We give a Pfaffian formula to compute the partition function of the Ising model on any graph GG embedded in a closed, possibly non-orientable surface. This formula, which is suitable for computational purposes, is based on the relation between the Ising model on GG and the dimer model on its terminal graph GTG^T. By combining the ideas of Loebl-Masbaum \cite{Loeb11}, Tesler \cite{Tes2000}, Cimasoni \cite{Cim09, Cim10} and Chelkak-Cimasoni-Kassel \cite{Chel15}, we give an elementary proof for the formula.

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